Stability and instability of a one-dimensional MHD model

TL;DR

This paper investigates the stability and instability of a one-dimensional MHD model, extending previous results from the De Gregorio model, and establishes well-posedness and instability results in various function spaces.

Contribution

It generalizes stability analysis from the De Gregorio model to a broader MHD model, identifying the primary instability mechanism and establishing well-posedness results.

Findings

Global well-posedness of the linearized system

Instability for a broad class of initial data

Stability in a specific subspace of the weighted Sobolev space

Abstract

We consider a one-dimensional magnetohydrodynamics model introduced by Dai \textit{et al.}~(2023), in a parameter regime where, in the absence of a magnetic field, the system reduces to the De Gregorio model for the Euler equations. We analyze stability and instability near the first excited state on the torus, thus generalizing the recent results obtained by Guo and Jiu~(2025) for the De Gregorio model. Specifically, we establish global well-posedness of the linearized system, local well-posedness for the nonlinear system, and demonstrate both linear and nonlinear instability for a broad class of initial data in the weighted Sobolev space introduced by Lai \textit{et al.}~(2020). We identify the principal linearized operator, which is structurally equivalent to that of the De Gregorio model, as the primary mechanism of instability. Moreover, we prove global well-posedness and stability of both linear and nonlinear systems for initial data in a particular subspace of the aforementioned weighted Sobolev space.